nLab rough paths

Idea

Rough Paths are a way of defining Riemann-Stieltjes type integrals for paths with poor $p$-variation where Ito techniques don’t work in the context of solving controlled differential equations.

Set-Up

We want to solve differential equations of the form:

$$y_t=y_0+\int_0^t V(y_s)\otimes dX_s$$

where $V=(V_1,...,V_d)$ with $V_i\in C_b^\infty(\mathbb{R}^d,\mathbb{R}^d)$ smooth bounded vector fields and $X\in C^p([0,T],\mathbb{R}^d)=\{f\colon \|f\|_{p-var}\colon = (\sup_{\mathcal{P}} |f_t-f_s|^p)^{1/p}\lt \infty\}$ is a signal with finite $p$ variation.

We want to solve this by a fixed point argument/Picard iteration. That means we must define:

$$\int_0^t f(X_s)\otimes dX_s$$

for smooth bounded $f$.

The classical paper (Young 1936) says, in short, that one can define this integral as a Riemann-Stieltjes integral iff $p\lt 2$.

A theorem by Bichdeller-Dellecherie says, in short, that one can define this integral in Itô sense? if and only if $X$ is a semimartingale?.

Main Result

Given a sequence of partitions, $\mathcal{P}_n=\{0=t_0\lt...\lt t_{n-1}\lt t\}$, if one can define $X^1_{su}\colon = \int_s^u dX_{r_1}$, $X^2_{su}\colon =\int_s^u\int_s^{r_2} dX_{r_1}\otimes dX_{r_2}$,…, $X^\ell_{su}\colon =\int_s^u \int_s^{r_\ell}...\int_s^{r_2} dX_{r_1}\otimes...\otimes dX_{r_\ell}$ where $\ell=\lfloor p \rfloor$ for $s,u\in[0,t]$ then the following “corrected Riemann sum” converges almost surely as is called the rough integral:

$$\lim_{n\to\infty}\sum_{k=0}^{n} f(X_{t_k} )X^1_{t_k t_{k+1}}+Df(X_{t_k})X^2_{t_k t_{k+1}}+...+D^{\ell-1}f(X_{t_k})X^{\ell}_{t_k t_{k+1}}\colon = \int_0^t f(X_S)\otimes d\mathbf{X}_s$$

The collection $\mathbf{X}=(X^1,...,X^\ell)$ is called a rough path of order $\ell$. We say that $(f(X),Df(X),...,D^{\ell-1}f(X))$ is controlled by $\mathbf{X}$.

Example

Fractional Brownian motion (fBm), $B_t$, is a centered, continuous Gaussian process with covariance

$$E(B_t B_s)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right)$$

where $H\in(0,1)$ is called the Hurst parameter. When $H=1/2$ we recover regular Brownian motion.

We have that $B_t\in C^p$ iff $p\gt 1/H$ and that fBm is not a semimartingale unless $H=1/2$. So SDEs driven by fBm with $H\lt 1/2$ are not solvable by classical means and rough paths must be used.

References

• LC Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), 251–282. doi:10.1007/BF02401743, (Project Euclid).

• Peter Friz and Martin Hairer?, A Course on Rough Paths, (2014), doi:10.1007/978-3-319-08332-2, (pdf).

• Peter Friz and Nicolas Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010) doi:10.1017/CBO9780511845079 (pdf).

• Ilya Chevyrev, Rough path theory, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2402.10331]